Integrand size = 22, antiderivative size = 424 \[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\frac {b (4 b c-5 a d) x}{10 a c (b c+a d)^2 \left (a-b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2+52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c+a d)^3 \sqrt [4]{a-b x^2}}+\frac {d x}{2 c (b c+a d) \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac {\sqrt {b} \left (12 b^2 c^2+52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c (b c+a d)^3 \sqrt [4]{a-b x^2}}+\frac {\sqrt [4]{a} d^{3/2} (11 b c+2 a d) \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c+a d)^{7/2} x}-\frac {\sqrt [4]{a} d^{3/2} (11 b c+2 a d) \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c+a d)^{7/2} x} \] Output:
1/10*b*(-5*a*d+4*b*c)*x/a/c/(a*d+b*c)^2/(-b*x^2+a)^(5/4)+1/10*b*(-5*a^2*d^ 2+52*a*b*c*d+12*b^2*c^2)*x/a^2/c/(a*d+b*c)^3/(-b*x^2+a)^(1/4)+1/2*d*x/c/(a *d+b*c)/(-b*x^2+a)^(5/4)/(d*x^2+c)-1/10*b^(1/2)*(-5*a^2*d^2+52*a*b*c*d+12* b^2*c^2)*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/a^(1/2))),2^ (1/2))/a^(3/2)/c/(a*d+b*c)^3/(-b*x^2+a)^(1/4)+1/4*a^(1/4)*d^(3/2)*(2*a*d+1 1*b*c)*(b*x^2/a)^(1/2)*EllipticPi((-b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2 )/(a*d+b*c)^(1/2),I)/c/(a*d+b*c)^(7/2)/x-1/4*a^(1/4)*d^(3/2)*(2*a*d+11*b*c )*(b*x^2/a)^(1/2)*EllipticPi((-b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d +b*c)^(1/2),I)/c/(a*d+b*c)^(7/2)/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.73 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (b d \left (-12 b^2 c^2-52 a b c d+5 a^2 d^2\right ) x^2 \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (-6 a c \left (10 a^4 d^3-15 a^3 b d^2 \left (-2 c+d x^2\right )-6 b^4 c^2 x^2 \left (c+2 d x^2\right )+2 a b^3 c \left (5 c^2-5 c d x^2-26 d^2 x^4\right )+a^2 b^2 d \left (30 c^2+26 c d x^2+5 d^2 x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (5 a^4 d^3-10 a^3 b d^3 x^2-12 b^4 c^2 x^2 \left (c+d x^2\right )+4 a b^3 c \left (4 c^2-9 c d x^2-13 d^2 x^4\right )+a^2 b^2 d \left (56 c^2+56 c d x^2+5 d^2 x^4\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )-b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (-a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{60 a^2 c^2 (b c+a d)^3 \sqrt [4]{a-b x^2}} \] Input:
Integrate[1/((a - b*x^2)^(9/4)*(c + d*x^2)^2),x]
Output:
(x*(b*d*(-12*b^2*c^2 - 52*a*b*c*d + 5*a^2*d^2)*x^2*(1 - (b*x^2)/a)^(1/4)*A ppellF1[3/2, 1/4, 1, 5/2, (b*x^2)/a, -((d*x^2)/c)] + (6*c*(-6*a*c*(10*a^4* d^3 - 15*a^3*b*d^2*(-2*c + d*x^2) - 6*b^4*c^2*x^2*(c + 2*d*x^2) + 2*a*b^3* c*(5*c^2 - 5*c*d*x^2 - 26*d^2*x^4) + a^2*b^2*d*(30*c^2 + 26*c*d*x^2 + 5*d^ 2*x^4))*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, -((d*x^2)/c)] + x^2*(5*a^4*d ^3 - 10*a^3*b*d^3*x^2 - 12*b^4*c^2*x^2*(c + d*x^2) + 4*a*b^3*c*(4*c^2 - 9* c*d*x^2 - 13*d^2*x^4) + a^2*b^2*d*(56*c^2 + 56*c*d*x^2 + 5*d^2*x^4))*(4*a* d*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, -((d*x^2)/c)] - b*c*AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, -((d*x^2)/c)])))/((-a + b*x^2)*(c + d*x^2)*(6*a*c* AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, -((d*x^2)/c)] + x^2*(-4*a*d*AppellF1 [3/2, 1/4, 2, 5/2, (b*x^2)/a, -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/ 2, (b*x^2)/a, -((d*x^2)/c)])))))/(60*a^2*c^2*(b*c + a*d)^3*(a - b*x^2)^(1/ 4))
Time = 0.68 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {316, 27, 402, 27, 402, 27, 405, 227, 226, 310, 25, 993, 1542}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}-\frac {\int -\frac {-7 b d x^2+4 b c+2 a d}{2 \left (a-b x^2\right )^{9/4} \left (d x^2+c\right )}dx}{2 c (a d+b c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 (2 b c+a d)-7 b d x^2}{\left (a-b x^2\right )^{9/4} \left (d x^2+c\right )}dx}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {2 \int \frac {12 b^2 c^2+40 a b d c+10 a^2 d^2+3 b d (4 b c-5 a d) x^2}{2 \left (a-b x^2\right )^{5/4} \left (d x^2+c\right )}dx}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 b d (4 b c-5 a d) x^2+2 \left (6 b^2 c^2+20 a b d c+5 a^2 d^2\right )}{\left (a-b x^2\right )^{5/4} \left (d x^2+c\right )}dx}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {12 b^3 c^3+52 a b^2 d c^2-60 a^2 b d^2 c-10 a^3 d^3+b d \left (12 b^2 c^2+52 a b d c-5 a^2 d^2\right ) x^2}{2 \sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{a (a d+b c)}+\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\int \frac {b d \left (12 b^2 c^2+52 a b d c-5 a^2 d^2\right ) x^2+2 \left (6 b^3 c^3+26 a b^2 d c^2-30 a^2 b d^2 c-5 a^3 d^3\right )}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {b \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt [4]{a-b x^2}}dx-5 a^2 d^2 (2 a d+11 b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 227 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {b \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{\sqrt [4]{a-b x^2}}-5 a^2 d^2 (2 a d+11 b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 226 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}-5 a^2 d^2 (2 a d+11 b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 310 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}-\frac {10 a^2 d^2 \sqrt {\frac {b x^2}{a}} (2 a d+11 b c) \int -\frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {a-b x^2}{a}} \left (b c+a d-d \left (a-b x^2\right )\right )}d\sqrt [4]{a-b x^2}}{x}}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {10 a^2 d^2 \sqrt {\frac {b x^2}{a}} (2 a d+11 b c) \int \frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {a-b x^2}{a}} \left (b c+a d-d \left (a-b x^2\right )\right )}d\sqrt [4]{a-b x^2}}{x}+\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}-\frac {10 a^2 d^2 \sqrt {\frac {b x^2}{a}} (2 a d+11 b c) \left (\frac {\int \frac {1}{\left (\sqrt {b c+a d}+\sqrt {d} \sqrt {a-b x^2}\right ) \sqrt {1-\frac {a-b x^2}{a}}}d\sqrt [4]{a-b x^2}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {b c+a d}-\sqrt {d} \sqrt {a-b x^2}\right ) \sqrt {1-\frac {a-b x^2}{a}}}d\sqrt [4]{a-b x^2}}{2 \sqrt {d}}\right )}{x}}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right )}{a \sqrt [4]{a-b x^2} (a d+b c)}-\frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} \left (-5 a^2 d^2+52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}-\frac {10 a^2 d^2 \sqrt {\frac {b x^2}{a}} (2 a d+11 b c) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d+b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d+b c}}\right )}{x}}{a (a d+b c)}}{5 a (a d+b c)}+\frac {2 b x (4 b c-5 a d)}{5 a \left (a-b x^2\right )^{5/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{5/4} \left (c+d x^2\right ) (a d+b c)}\) |
Input:
Int[1/((a - b*x^2)^(9/4)*(c + d*x^2)^2),x]
Output:
(d*x)/(2*c*(b*c + a*d)*(a - b*x^2)^(5/4)*(c + d*x^2)) + ((2*b*(4*b*c - 5*a *d)*x)/(5*a*(b*c + a*d)*(a - b*x^2)^(5/4)) + ((2*b*(12*b^2*c^2 + 52*a*b*c* d - 5*a^2*d^2)*x)/(a*(b*c + a*d)*(a - b*x^2)^(1/4)) - ((2*Sqrt[a]*Sqrt[b]* (12*b^2*c^2 + 52*a*b*c*d - 5*a^2*d^2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcS in[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(a - b*x^2)^(1/4) - (10*a^2*d^2*(11*b*c + 2 *a*d)*Sqrt[(b*x^2)/a]*((a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[b*c + a*d]), ArcSin[(a - b*x^2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[b*c + a*d]) - (a^(1/4)*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[b*c + a*d], ArcSin[(a - b*x^ 2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[b*c + a*d])))/x)/(a*(b*c + a*d)))/ (5*a*(b*c + a*d)))/(4*c*(b*c + a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[2*(Sqrt[(-b)*(x^2/a)]/x) Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 ), x_Symbol] :> Simp[f/d Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
\[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {9}{4}} \left (x^{2} d +c \right )^{2}}d x\]
Input:
int(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x)
Output:
int(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x)
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a - b x^{2}\right )^{\frac {9}{4}} \left (c + d x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(-b*x**2+a)**(9/4)/(d*x**2+c)**2,x)
Output:
Integral(1/((a - b*x**2)**(9/4)*(c + d*x**2)**2), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(9/4)*(d*x^2 + c)^2), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(9/4)*(d*x^2 + c)^2), x)
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{9/4}\,{\left (d\,x^2+c\right )}^2} \,d x \] Input:
int(1/((a - b*x^2)^(9/4)*(c + d*x^2)^2),x)
Output:
int(1/((a - b*x^2)^(9/4)*(c + d*x^2)^2), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} c^{2}+2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} c d \,x^{2}+\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} d^{2} x^{4}-2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a b \,c^{2} x^{2}-4 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a b c d \,x^{4}-2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a b \,d^{2} x^{6}+\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} c^{2} x^{4}+2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} c d \,x^{6}+\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} d^{2} x^{8}}d x \] Input:
int(1/(-b*x^2+a)^(9/4)/(d*x^2+c)^2,x)
Output:
int(1/((a - b*x**2)**(1/4)*a**2*c**2 + 2*(a - b*x**2)**(1/4)*a**2*c*d*x**2 + (a - b*x**2)**(1/4)*a**2*d**2*x**4 - 2*(a - b*x**2)**(1/4)*a*b*c**2*x** 2 - 4*(a - b*x**2)**(1/4)*a*b*c*d*x**4 - 2*(a - b*x**2)**(1/4)*a*b*d**2*x* *6 + (a - b*x**2)**(1/4)*b**2*c**2*x**4 + 2*(a - b*x**2)**(1/4)*b**2*c*d*x **6 + (a - b*x**2)**(1/4)*b**2*d**2*x**8),x)